Physica D: Nonlinear Phenomena [SJR: 1.048] [H-I: 89] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0167-2789 Published by Elsevier [2801 journals] |
- Whitham modulation equations, coalescing characteristics, and dispersive
Boussinesq dynamics- Abstract: Publication date: Available online 21 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Daniel J. Ratliff, Thomas J. Bridges
Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1 + 1 and 2 + 1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and don’t depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein Gordon equation is used to illustrate the theory.
PubDate: 2016-01-25T10:24:58Z
- Abstract: Publication date: Available online 21 January 2016
- The propagation of internal undular bores over variable topography
- Abstract: Publication date: Available online 25 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): R. Grimshaw, C. Yuan
In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg-de Vries type. Typically they occur in regions of variable bottom topography when the variable-coefficient Korteweg-de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly.
PubDate: 2016-01-25T10:24:58Z
- Abstract: Publication date: Available online 25 January 2016
- Observation of dispersive shock waves developing from initial depressions
in shallow water- Abstract: Publication date: Available online 25 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): S. Trillo, M. Klein, G. Clauss, M. Onorato
We investigate surface gravity waves in a shallow water tank, in the limit of long wavelengths. We report the observation of non-stationary dispersive shock waves rapidly expanding over a 90 m flume. They are excited by means of a wave maker that allows us to launch a controlled smooth (single well) depression with respect to the unperturbed surface of the still water, a case that contains no solitons. The dynamics of the shock waves are observed at different levels of nonlinearity equivalent to a different relative smallness of the dispersive effect. The observed undulatory behaviour is found to be in good agreement with the dynamics described in terms of a Korteweg-de Vries equation with evolution in space, though in the most nonlinear cases the description turns out to be improved over the quasi linear trailing edge of the shock by modelling the evolution in terms of the integro-differential (nonlocal) Whitham equation.
PubDate: 2016-01-25T10:24:58Z
- Abstract: Publication date: Available online 25 January 2016
- Sine-Gordon modulation solutions: Application to macroscopic non-lubricant
friction- Abstract: Publication date: Available online 19 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Naum I. Gershenzon, Gust Bambakidis, Thomas E. Skinner
The Frenkel-Kontorova (FK) model and its continuum approximation, the sine-Gordon (SG) equation, are widely used to model a variety of important nonlinear physical systems. Many practical applications require the wave-train solution, which includes many solitons. In such cases, an important and relevant extension of these models applies Whitham’s averaging procedure to the SG equation. The resulting SG modulation equations describe the behavior of important measureable system parameters that are the average of the small-scale solutions given by the SG equation. A fundamental problem of modern physics that is the topic of this paper is the description of the transitional process from a static to a dynamic frictional regime. We have shown that the SG modulation equations are a suitable apparatus for describing this transition. The model provides relations between kinematic (rupture and slip velocities) and dynamic (shear and normal stresses) parameters of the transition process. A particular advantage of the model is its ability to describe frictional processes over a wide range of rupture and slip velocities covering seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities on the order of a few km/day.
PubDate: 2016-01-21T10:08:24Z
- Abstract: Publication date: Available online 19 January 2016
- Rigorous numerics for NLS: Bound states, spectra, and controllability
- Abstract: Publication date: Available online 21 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Roberto Castelli, Holger Teismann
In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound–state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof (Beauchard et al., 2015) of the local exact controllability of NLS.
PubDate: 2016-01-21T10:08:24Z
- Abstract: Publication date: Available online 21 January 2016
- Reaction–diffusion–advection approach to spatially localized
treadmilling aggregates of molecular motors- Abstract: Publication date: Available online 12 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Arik Yochelis, Tomer Bar-On, Nir S. Gov
Unconventional myosins belong to a class of molecular motors that walk processively inside cellular protrusions towards the tips, on top of actin filament. Surprisingly, in addition, they also form retrograde moving self-organized aggregates. The quantitative properties of these aggregates are recapitulated by a mass conserving reaction–diffusion–advection model and admit two distinct families of modes: traveling waves and pulse trains. Unlike the traveling waves that are generated by a linear instability, pulses are nonlinear structures that propagate on top of linearly stable uniform backgrounds. Asymptotic analysis of isolated pulses via a simplified reaction–diffusion–advection variant on large periodic domains, allows to draw qualitative trends for pulse properties, such as the amplitude, width, and propagation speed. The results agree well with numerical integrations and related to available empirical observations.
PubDate: 2016-01-17T10:00:08Z
- Abstract: Publication date: Available online 12 January 2016
- Autoresonance versus localization in weakly coupled oscillators
- Abstract: Publication date: Available online 8 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Agnessa Kovaleva, Leonid I. Manevitch
We study formation of autoresonance (AR) in a two-degree of freedom oscillator array including a nonlinear (Duffing) oscillator (the actuator) weakly coupled to a linear attachment. Two classes of systems are studied. In the first class of systems, a periodic force with constant (resonance) frequency is applied to a nonlinear oscillator (actuator) with slowly time-decreasing stiffness. In the systems of the second class a nonlinear time-invariant oscillator is subjected to an excitation with slowly increasing frequency. In both cases, the attached linear oscillator and linear coupling are time-invariant, and the system is initially engaged in resonance. This paper demonstrates that in the systems of the first type AR in the nonlinear actuator entails oscillations with growing amplitudes in the linear attachment while in the system of the second type energy transfer from the nonlinear actuator is insufficient to excite high-energy oscillations of the attachment. It is also shown that a slow change of stiffness may enhance the response of the actuator and make it sufficient to support oscillations with growing energy in the attachment even beyond the linear resonance. Explicit asymptotic approximations of the solutions are obtained. Close proximity of the derived approximations to exact (numerical) results is demonstrated.
PubDate: 2016-01-12T00:45:22Z
- Abstract: Publication date: Available online 8 January 2016
- Actomyosin contraction, aggregation and traveling waves in a treadmilling
actin array- Abstract: Publication date: Available online 4 January 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Dietmar Oelz, Alex Mogilner
We use perturbation theory to derive a continuum model for the dynamic actomyosin bundle/ring in the regime of very strong crosslinking. Actin treadmilling is essential for contraction. Linear stability analysis and numerical solutions of the model equations reveal that when the actin treadmilling is very slow, actin and myosin aggregate into equidistantly spaced peaks. When treadmilling is significant, actin filament of one polarity are distributed evenly, while filaments of the opposite polarity develop a shock wave moving with the treadmilling velocity. Myosin aggregates into a sharp peak surfing the crest of the actin wave. Any actomyosin aggregation diminishes contractile stress. The easiest way to maintain higher contraction is to upregulate the actomyosin turnover which destabilizes nontrivial patterns and stabilizes the homogeneous actomyosin distributions. We discuss the model’s implications for the experiment.
PubDate: 2016-01-08T00:29:05Z
- Abstract: Publication date: Available online 4 January 2016
- Metastable switching in a planar limit cycle system with additive noise
- Abstract: Publication date: Available online 24 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Michael A. Schwemmer, Jay M. Newby
Additive noise is known to produce counter-intuitive behaviors in nonlinear dynamical systems. Previously, it was shown that systems with a deterministic limit cycle can display switching between metastable states in the presence of asymmetric additive white noise. Here, we systematically analyze the dynamics of this metastable behavior and show how the vector field away from the limit cycle influences the rate and directionality of the metastable switching. Using stochastic phase reduction methods as well as asymptotic approximations, we identify mechanisms underlying different rates of switching and predict when the system will rotate in the opposite direction of the deterministic limit cycle. Thus, this work presents an alternative mechanism for generating a range of metastable switch behaviors that have been observed in a number of physical systems.
PubDate: 2016-01-04T00:20:31Z
- Abstract: Publication date: Available online 24 December 2015
- Nonlinear disintegration of sine wave in the framework of the Gardner
equation- Abstract: Publication date: Available online 29 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Oxana Kurkina, Ekaterina Rouvinskaya, Tatiana Talipova, Andrey Kurkin, Efim Pelinovsky
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg–de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg–de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of “exchange” or “overtake” types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when “free” velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
PubDate: 2016-01-04T00:20:31Z
- Abstract: Publication date: Available online 29 December 2015
- Symmetry types and phase-shift synchrony in networks
- Abstract: Publication date: Available online 31 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Martin Golubitsky, Leopold Matamba Messi, Lucy E. Spardy
In this paper we discuss what is known about the classification of symmetry groups and patterns of phase-shift synchrony for periodic solutions of coupled cell networks. Specifically, we compare the lists of spatial and spatiotemporal symmetries of periodic solutions of admissible vector fields to those of equivariant vector fields in the three cases of R n (coupled equations), T n (coupled oscillators), and ( R k ) n where k ≥ 2 (coupled systems). To do this we use the H / K Theorem of Buono and Golubitsky (2001) applied to coupled equations and coupled systems and prove the H / K theorem in the case of coupled oscillators. Josić and Török (2006) prove that the H / K lists for equivariant vector fields and admissible vector fields are the same for transitive coupled systems. We show that the corresponding theorem is false for coupled equations. We also prove that the pairs of subgroups H ⊃ K for coupled equations are contained in the pairs for coupled oscillators which are contained in the pairs for coupled systems. Finally, we prove that patterns of rigid phase-shift synchrony for coupled equations are contained in those of coupled oscillators and those of coupled systems.
PubDate: 2016-01-04T00:20:31Z
- Abstract: Publication date: Available online 31 December 2015
- Discrete synchronization of massively connected systems using hierarchical
couplings- Abstract: Publication date: Available online 23 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Camille Poignard
We study the synchronization of massively connected dynamical systems for which the interactions come from the succession of couplings forming a global hierarchical coupling process. Motivations of this work come from the growing necessity of understanding properties of complex systems that often exhibit a hierarchical structure. Starting with a set of 2 n systems, the couplings we consider represent a two-by-two matching process that gather them in larger and larger groups of systems, providing to the whole set a structure in n stages, corresponding to n scales of hierarchy. This leads us naturally to the synchronization of a Cantor set of systems, indexed by { 0 , 1 } N , using the closed-open sets defined by n -tuples of 0 and 1 that permit us to make the link with the finite previous situation of 2 n systems: we obtain a global synchronization result generalizing this case. In the same context, we deal with this question when some defects appear in the hierarchy, that is to say when some couplings among certain systems do not happen at a given stage of the hierarchy. We prove we can accept an infinite number of broken links inside the hierarchy while keeping a local synchronization, under the condition that these defects are present at the N smallest scales of the hierarchy (for a fixed integer N ) and they be enough spaced out in those scales.
PubDate: 2016-01-04T00:20:31Z
- Abstract: Publication date: Available online 23 December 2015
- Halo orbits around the collinear points of the restricted three-body
problem- Abstract: Publication date: Available online 23 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Marta Ceccaroni, Alessandra Celletti, Giuseppe Pucacco
We perform an analytical study of the bifurcation of the halo orbits around the collinear points L 1 , L 2 , L 3 for the circular, spatial, restricted three–body problem. Following a standard procedure, we reduce to the center manifold constructing a normal form adapted to the synchronous resonance. Introducing a detuning, which measures the displacement from the resonance and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio. In most cases, the analytical results thus obtained are in very good agreement with the numerical expectations, providing the bifurcation threshold with good accuracy. Care must be taken when dealing with L 3 for small values of the mass-ratio between the primaries; in that case, the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold.
PubDate: 2016-01-04T00:20:31Z
- Abstract: Publication date: Available online 23 December 2015
- Traveling waves for a model of gravity-driven film flows in cylindrical
domains- Abstract: Publication date: Available online 19 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Roberto Camassa, Jeremy L. Marzuola, H. Reed Ogrosky, Nathan Vaughn
Traveling wave solutions are studied for a recently-derived model of a falling viscous film on the interior of a vertical rigid tube. By identifying a Hopf bifurcation and using numerical continuation software, families of non-trivial traveling wave solutions may be traced out in parameter space. These families all contain a single solution at a ‘turnaround point’ with larger film thickness than all others in the family. In an earlier paper, it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model. Here, these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques. The linear stability of these solutions is also discussed; both large- and small-amplitude solutions are shown to be unstable, though the instability mechanisms are different for each wave type. Specifically, for small-amplitude waves, the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat-film solution unstable; for large-amplitude waves, this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small-amplitude waves.
PubDate: 2015-12-20T13:57:14Z
- Abstract: Publication date: Available online 19 December 2015
- Continuation of point clouds via persistence diagrams
- Abstract: Publication date: Available online 19 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Marcio Gameiro, Yasuaki Hiraoka, Ippei Obayashi
In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton-Raphson continuation method in this setting. Given an original point cloud P , its persistence diagram D , and a target persistence diagram D ′ , we gradually move from D to D ′ , by successively computing intermediate point clouds until we finally find a point cloud P ′ having D ′ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.
PubDate: 2015-12-20T13:57:14Z
- Abstract: Publication date: Available online 19 December 2015
- Orbital stability of periodic traveling-wave solutions for the regularized
Schamel equation- Abstract: Publication date: Available online 14 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Thiago Pinguello de Andrade, Ademir Pastor
In this work we study the orbital stability of periodic traveling-wave solutions for dispersive models. The study of traveling waves started in the mid-18th century when John S. Russel established that the flow of water waves in a shallow channel has constant evolution. In recent years, the general strategy to obtain orbital stability consists in proving that the traveling wave in question minimizes a conserved functional restricted to a certain manifold. Although our method can be applied to other models, we deal with the regularized Schamel equation, which contains a fractional nonlinear term. We obtain a smooth curve of periodic traveling-wave solutions depending on the Jacobian elliptic functions and prove that such solutions are orbitally stable in the energy space. In our context, instead of minimizing the augmented Hamiltonian in the natural codimension two manifold, we minimize it in a “new” manifold, which is suitable to our purposes.
PubDate: 2015-12-15T13:41:38Z
- Abstract: Publication date: Available online 14 December 2015
- Horizontal rolls over localized heat source in a cylindrical layer
- Abstract: Publication date: Available online 2 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): A. Sukhanovskii, A. Evgrafova, E. Popova
Convection over a localized heat source in a cylindrical layer was studied experimentally and numerically for fluids with different values of Prandtl number. A basic flow produced by a horizontal temperature gradient occupies the whole layer and leads to unstable temperature stratification over the heating area and the formation of a complex system of secondary flows. The main focus of our study was the spatial and temporal evolution of small-scale convective structures in the boundary layer of a basic flow. Transitions from transverse rolls to radial rolls and further to their superposition were found in experiment and numerical simulation. Various types of visualization revealed co-existence of different kinds of secondary flows. Complex convective patterns over the heating area are temporally periodic. The characteristic frequency of transverse rolls depends on Rayleigh number for a wide range of governing parameters.
PubDate: 2015-12-06T13:25:28Z
- Abstract: Publication date: Available online 2 December 2015
- Phenomenological model for predicting stationary and non-stationary
spectra of wave turbulence in vibrating plates- Abstract: Publication date: Available online 1 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): T. Humbert, C. Josserand, C. Touzé, O. Cadot
A phenomenological model describing the time-frequency dependence of the power spectrum of thin plates vibrating in a wave turbulence regime, is introduced. The model equation contains as basic solutions the Rayleigh-Jeans equipartition of energy, as well as the Kolmogorov-Zakharov spectrum of wave turbulence. In the Wave Turbulence Theory framework, the model is used to investigate the self-similar, non-stationary solutions of forced and free turbulent vibrations. Frequency-dependent damping laws can easily be accounted for. Their effects on the characteristics of the stationary spectra of turbulence are then investigated. Thanks to this analysis, self-similar universal solutions are given, relating the power spectrum to both the injected power and the damping law.
PubDate: 2015-12-06T13:25:28Z
- Abstract: Publication date: Available online 1 December 2015
- Whitham theory for perturbed Korteweg-de Vries equation
- Abstract: Publication date: Available online 4 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): A.M. Kamchatnov
Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg-de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained in both cases. The essential difference between them is illustrated by an example of so-called ‘generalized Korteweg-de Vries equation’. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.
PubDate: 2015-12-06T13:25:28Z
- Abstract: Publication date: Available online 4 December 2015
- Dynamical decision making in a genetic perceptron
- Abstract: Publication date: Available online 2 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Svetlana Filicheva, Alexey Zaikin, Oleg Kanakov
Decision making is an essential element of cell functioning, which determines milestones of its evolution including differentiation, apoptosis and possible transition to cancerous state. Recently the concept of stochastic resonance in decision making (SRIDM) was introduced, demonstrated and explained using a synthetic genetic classifier circuit as an example. It manifests itself as a maximum in the dependence of classification accuracy upon noise intensity, and was caused by the concurrent action of two factors, both coarsening the classification accuracy by themselves, but found to extenuate the effect of each other: perturbation of classifier threshold and additive noise in classifier inputs. In the present work we extend the SRIDM concept to dynamical decision making, in which a classifier keeps track of the changeable input. We reproduce the stochastic resonance effect caused by noise and threshold perturbation, and demonstrate a new mechanism of SRIDM, which is associated with bistability and not connected with threshold perturbation.
PubDate: 2015-12-06T13:25:28Z
- Abstract: Publication date: Available online 2 December 2015
- Collapse for the higher-order nonlinear Schrödinger equation
- Abstract: Publication date: Available online 1 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): V. Achilleos, S. Diamantidis, D.J. Frantzeskakis, T.P. Horikis, N.I. Karachalios, P.G. Kevrekidis
We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schrödinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.
PubDate: 2015-12-01T13:03:15Z
- Abstract: Publication date: Available online 1 December 2015
- Lossless polariton solitons
- Abstract: Publication date: Available online 1 December 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Stavros Komineas, Stephen P. Shipman, Stephanos Venakides
Photons and excitons in a semiconductor microcavity interact to form exciton-polariton condensates. These are governed by a nonlinear quantum-mechanical system involving exciton and photon wavefunctions. We calculate all non-traveling harmonic soliton solutions for the one-dimensional lossless system. There are two frequency bands of bright solitons when the inter-exciton interactions produce an attractive nonlinearity and two frequency bands of dark solitons when the nonlinearity is repulsive. In addition, there are two frequency bands for which the exciton wavefunction is discontinuous at its symmetry point, where it undergoes a phase jump of π . A band of continuous dark solitons merges with a band of discontinuous dark solitons, forming a larger band over which the soliton far-field amplitude varies from 0 to ∞ ; the discontinuity is initiated when the operating frequency exceeds the free exciton frequency. The far fields of the solitons in the lowest and highest frequency bands (one discontinuous and one continuous dark) are linearly unstable, whereas the other four bands have linearly stable far fields, including the merged band of dark solitons.
PubDate: 2015-12-01T13:03:15Z
- Abstract: Publication date: Available online 1 December 2015
- Vortex nucleation in a dissipative variant of the nonlinear
Schrödinger equation under rotation- Abstract: Publication date: Available online 27 November 2015
Source:Physica D: Nonlinear Phenomena
Author(s): R. Carretero-González, P.G. Kevrekidis, T. Kolokolnikov
In the present work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schrödinger equation under the impact of external rotation. As in the well established Hamiltonian case, the rotation gives rise to the formation of vortices. We show, however, that the most unstable mode leading to this instability scales with an appropriate power of the chemical potential μ of the system, increasing proportionally to μ 2 / 3 . The precise form of the relevant formula, obtained through our asymptotic analysis, provides the most unstable mode as a function of the atomic density and the trap strength. We show how these unstable modes typically nucleate a large number of vortices in the periphery of the atomic cloud. However, through a pattern selection mechanism, prompted by symmetry-breaking, only few isolated vortices are pulled in sequentially from the periphery towards the bulk of the cloud resulting in highly symmetric stable vortex configurations with far fewer vortices than the original unstable mode. These results may be of relevance to the experimentally tractable realm of finite temperature atomic condensates.
PubDate: 2015-11-27T12:49:00Z
- Abstract: Publication date: Available online 27 November 2015
- Stirring a fluid at low Reynolds numbers: Hydrodynamic collective effects
of active proteins in biological cells- Abstract: Publication date: Available online 14 November 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Raymond Kapral, Alexander S. Mikhailov
Most of the proteins in the cell, including not only molecular motors and machines, but also enzymes, are active. When ATP or other substrates are supplied, these macromolecules cyclically change their conformations. Therefore, they mechanically stir the cytoplasm and nucleoplasm, so that non-thermal fluctuating flows are produced. As we have recently shown [PNAS 112, E3639 (2015)], stochastic advection by such flows might lead to substantial diffusion enhancement of particles inside a living cell. Additionally, when gradients in the concentrations of active particles or in the ATP/substrate supply are present, chemotaxis-like drift should take place. Here, the motion of passive tracers with various sizes in a mixture of different kinds of active proteins is analyzed. Moreover, effects of hydrodynamic interactions on the motion of active proteins are explored. Theoretical results are compared with available experimental data for ATP-dependent diffusion of natural and microinjected particles in biological cells.
PubDate: 2015-11-18T12:19:23Z
- Abstract: Publication date: Available online 14 November 2015
- Contact-based model for strategy updating and evolution of cooperation
- Abstract: Publication date: Available online 12 November 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Jianlei Zhang, Zengqiang Chen
To establish an available model for the astoundingly strategy decision process of players is not easy, sparking heated debate about the related strategy updating rules is intriguing. Models for evolutionary games have traditionally assumed that players imitate their successful partners by the comparison of respective payoffs, raising the question of what happens if the game information is not easily available. Focusing on this yet-unsolved case, the motivation behind the work presented here is to establish a novel model for the updating of states in a spatial population, by detouring the required payoffs in previous models and considering much more players’ contact patterns. It can be handy and understandable to employ switching probabilities for determining the microscopic dynamics of strategy evolution. Our results illuminate the conditions under which the steady coexistence of competing strategies is possible. These findings reveal that the evolutionary fate of the coexisting strategies can be calculated analytically, and provide novel hints for the resolution of cooperative dilemmas in a competitive context. We hope that our results have disclosed new explanations about the survival and coexistence of competing strategies in structured populations.
PubDate: 2015-11-14T12:15:33Z
- Abstract: Publication date: Available online 12 November 2015
- On degree-degree correlations in multilayer networks
- Abstract: Publication date: Available online 12 November 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Guilherme Ferraz de Arruda, Emanuele Cozzo, Yamir Moreno, Francisco A. Rodrigues
We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree-degree correlations on multilayer systems, instead of on aggregated networks.
PubDate: 2015-11-14T12:15:33Z
- Abstract: Publication date: Available online 12 November 2015
- Erratum to “The Kuramoto model of coupled oscillators with a
bi-harmonic coupling function” [Physica D 289 (2014) 18–31]- Abstract: Publication date: 1 December 2015
Source:Physica D: Nonlinear Phenomena, Volume 313
Author(s): M. Komarov, A. Pikovsky
PubDate: 2015-11-10T11:59:12Z
- Abstract: Publication date: 1 December 2015
- Uniform modeling of bacterial colony patterns with varying nutrient and
substrate- Abstract: Publication date: Available online 10 November 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Deborah Schwarcz, Herbert Levine, Eshel Ben-Jacob, Gil Ariel
Bacteria develop complex patterns depending on growth condition. For example, Bacillus subtilisexhibit five different patterns depending on substrate hardness and nutrient concentration. We present a unified integro-differential model that reproduces the entire experimentally observed morphology diagram at varying nutrient concentrations and substrate hardness. The model allows a comprehensive and quantitative comparison between experimental and numerical variables and parameters, such as colony growth rate, nutrient concentration and diffusion constants. As a result, the role of the different physical mechanisms underlying and regulating the growth of the colony can be evaluated.
PubDate: 2015-11-10T11:59:12Z
- Abstract: Publication date: Available online 10 November 2015
- Simple nonlinear models suggest variable star universality
- Abstract: Publication date: Available online 10 November 2015
Source:Physica D: Nonlinear Phenomena
Author(s): John F. Lindner, Vivek Kohar, Behnam Kia, Michael Hippke, John G. Learned, William L. Ditto
Dramatically improved data from observatories like the CoRoT and Kepler spacecraft have recently facilitated nonlinear time series analysis and phenomenological modeling of variable stars, including the search for strange (aka fractal) or chaotic dynamics. We recently argued [Lindner et al., Phys. Rev. Lett. 114 (2015) 054101] that the Kepler data includes “golden” stars, whose luminosities vary quasiperiodically with two frequencies nearly in the golden ratio, and whose secondary frequencies exhibit power-law scaling with exponent near − 1.5 , suggesting strange nonchaotic dynamics and singular spectra. Here we use a series of phenomenological models to make plausible the connection between golden stars and fractal spectra. We thereby suggest that at least some features of variable star dynamics reflect universal nonlinear phenomena common to even simple systems.
PubDate: 2015-11-10T11:59:12Z
- Abstract: Publication date: Available online 10 November 2015
- Editorial Board
- Abstract: Publication date: 1 December 2015
Source:Physica D: Nonlinear Phenomena, Volume 313
PubDate: 2015-11-10T11:59:12Z
- Abstract: Publication date: 1 December 2015
- Polar rotation angle identifies elliptic islands in unsteady dynamical
systems- Abstract: Publication date: 1 February 2016
Source:Physica D: Nonlinear Phenomena, Volume 315
Author(s): Mohammad Farazmand, George Haller
We propose rotation inferred from the polar decomposition of the flow gradient as a diagnostic for elliptic (or vortex-type) invariant regions in non-autonomous dynamical systems. We consider here two- and three-dimensional systems, in which polar rotation can be characterized by a single angle. For this polar rotation angle (PRA), we derive explicit formulas using the singular values and vectors of the flow gradient. We find that closed level sets of the PRA reveal elliptic islands in great detail, and singular level sets of the PRA uncover centers of such islands. Both features turn out to be objective (frame-invariant) for two-dimensional systems. We illustrate the diagnostic power of PRA for elliptic structures on several examples.
PubDate: 2015-11-06T11:46:42Z
- Abstract: Publication date: 1 February 2016
- High-order control for symplectic maps
- Abstract: Publication date: Available online 30 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): M. Sansottera, A. Giorgilli, T. Carletti
We revisit the problem of introducing an a priori control for devices that can be modeled via a symplectic map in a neighborhood of an elliptic equilibrium. Using a technique based on Lie transform methods we produce a normal form algorithm that avoids the usual step of interpolating the map with a flow. The formal algorithm is completed with quantitative estimates that bring into evidence the asymptotic character of the normal form transformation. Then we perform an heuristic analysis of the dynamical behavior of the map using the invariant function for the normalized map. Finally, we discuss how control terms of different orders may be introduced so as to increase the size of the stable domain of the map. The numerical examples are worked out on a two dimensional map of Hénon type.
PubDate: 2015-11-02T05:44:31Z
- Abstract: Publication date: Available online 30 October 2015
- Minimal topological chaos coexisting with a finite set of homoclinic and
periodic orbits- Abstract: Publication date: Available online 24 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Valentín Mendoza, Walter Huaraca
In this note we explain how to find the minimal topological chaos relative to finite set of homoclinic and periodic orbits. The main tool is the pruning method, which is used for finding a hyperbolic map, obtained uncrossing pieces of the invariant manifolds, whose basic set contains all orbits forced by the finite set under consideration. Then we will show applications related to transport phenomena and to the problem of determining the orbits structure coexisting with a finite number of periodic orbits arising from the bouncing ball model.
PubDate: 2015-10-29T05:37:53Z
- Abstract: Publication date: Available online 24 October 2015
- Mixed dynamics in a parabolic standard map
- Abstract: Publication date: Available online 24 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): L.M. Lerman, J.D. Meiss
We use numerical and analytical tools to provide arguments in favor of the existence of a family of smooth, symplectic diffeomorphisms of the two-dimensional torus that have both a positive measure set with positive Lyapunov exponent and a positive measure set with zero Lyapunov exponent. The family we study is the unfolding of an almost-hyperbolic diffeomorphism on the boundary of the set of Anosov diffeomorphisms, proposed by Lewowicz.
PubDate: 2015-10-29T05:37:53Z
- Abstract: Publication date: Available online 24 October 2015
- Poincaré inverse problem and torus construction in phase space
- Abstract: Publication date: Available online 26 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Teemu Laakso, Mikko Kaasalainen
The phase space of an integrable Hamiltonian system is foliated by invariant tori. For an arbitrary Hamiltonian H such a foliation may not exist, but we can artificially construct one through a parameterised family of surfaces, with the intention of finding, in some sense, the closest integrable approximation to H . This is the Poincaré inverse problem (PIP). In this paper, we review the available methods of solving the PIP and present a new iterative approach which works well for the often problematic thin orbits.
PubDate: 2015-10-29T05:37:53Z
- Abstract: Publication date: Available online 26 October 2015
- Clustering of extreme events created by multiple correlated maxima
- Abstract: Publication date: Available online 22 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Davide Azevedo, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, Fagner B. Rodrigues
We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The novelty is that we will consider observables achieving a global maximum value (possible infinite) at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point. These multiple correlated maxima can be seen as a new mechanism creating clustering of extreme observations, i.e., the occurrence of several extreme observations concentrated in the time frame. We recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point. We will study this mechanism for creating clustering and will address the existence of limiting Extreme Value Laws, the repercussions on the value of the Extremal Index, the impact on the limit of Rare Events Points Processes, the influence on clustering patterns and the competition of domains of attraction. We also consider briefly and for comparison purposes multiple uncorrelated maxima. The systems considered include expanding maps of the interval such as Rychlik maps but also maps with an indifferent fixed point such as Manneville-Pommeau maps.
PubDate: 2015-10-25T05:31:47Z
- Abstract: Publication date: Available online 22 October 2015
- Partner orbits and action differences on compact factors of the hyperbolic
plane. II: Higher-order encounters- Abstract: Publication date: Available online 22 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Hien Minh Huynh
Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. We specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. The companion paper (Huynh and Kunze, 2015) proved the existence of a unique periodic partner orbit for a given periodic orbit with a small-angle self-crossing in configuration space that is a 2-encounter and derived an estimate for the action difference of the orbit pair. In this paper, we provide an inductive argument to deal with higher-order encounters: we prove that a given periodic orbit including an L -parallel encounter has ( L − 1 ) ! − 1 partner orbits; we construct partner orbits and give estimates for the action differences between orbit pairs.
PubDate: 2015-10-25T05:31:47Z
- Abstract: Publication date: Available online 22 October 2015
- Positive and necklace solitary waves on bounded domains
- Abstract: Publication date: Available online 13 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): G. Fibich, D. Shpigelman
We present new solitary wave solutions of the two-dimensional nonlinear Schrödinger equation on bounded domains (such as rectangles, circles, and annuli). These multi-peak “necklace” solitary waves consist of several identical positive profiles (“pearls”), such that adjacent “pearls” have opposite signs. They are stable at low powers, but become unstable at powers well below the critical power for collapse P cr . This is in contrast with the ground-state (“single-pearl”) solitary waves on bounded domains, which are stable at any power below P cr . On annular domains, the ground state solitary waves are radial at low powers, but undergo a symmetry breaking at a threshold power well below P cr . As in the case of convex bounded domains, necklace solitary waves on the annulus are stable at low powers and become unstable at powers well below P cr . Unlike on convex bounded domains, however, necklace solitary waves on the annulus have a second stability regime at powers well above P cr . For example, when the ratio of the inner to outer radii is 1:2, four-pearl necklaces are stable when their power is between 3.1 P cr and 3.7 P cr . This finding opens the possibility to propagate localized laser beams with substantially more power than was possible until now. The instability of necklace solitary waves is excited by perturbations that break the antisymmetry between adjacent pearls, and is manifested by power transfer between pearls. In particular, necklace instability is unrelated to collapse. In order to compute numerically the profile of necklace solitary waves on bounded domains, we introduce a non-spectral variant of Petviashvili’s renormalization method.
PubDate: 2015-10-20T12:15:53Z
- Abstract: Publication date: Available online 13 October 2015
- Isles within islets: The lattice origin of small-world networks in
pancreatic tissues- Abstract: Publication date: Available online 13 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Amlan K. Barua, Pranay Goel
The traditional computational model of the pancreatic islets of Langerhans is a lattice of β -cells connected with gap junctions. Numerous studies have investigated the behavior of networks of coupled β -cells and have shown that gap junctions synchronize bursting strongly. This simplistic architecture of islets, however, seems increasingly untenable at the face of recent experimental advances. In a microfluidics experiment on isolated islets, Rocheleau et al. (2004) showed a failure of penetration of excitation when one end received high glucose and other end was not excited sufficiently; this suggested that gap junctions may not be efficient at inducing synchrony throughout the islet. Recently, Stozer et al. (2013) have argued that the functional networks of β -cells in an islet are small world. Their results implicate the existence of a few long-range connections among cells in the network. The physiological reason underlying this claim is not well understood. These studies cast doubt on the original lattice model that largely predict an all-or-none synchrony among the cells. Here we have attempted to reconcile these observations in a unified framework. We assume that cells in the islet are coupled randomly to their nearest neighbors with some probability, p . We simulated detailed β -cell bursting in such islets. By varying p systematically we were led to network parameters similar to those obtained by Stozer et al. (2013). We find that the networks within islets break up into components giving rise to smaller isles within the super structure–isles-within-islets, as it were. This structure can also account for the partial excitation seen by Rocheleau et al. (2004) Our updated view of islet architecture thus explains the paradox how islets can have strongly synchronizing gap junctions, and be weakly coordinated at the same time.
PubDate: 2015-10-20T12:15:53Z
- Abstract: Publication date: Available online 13 October 2015
- Stochastic stability of measures in gradient systems
- Abstract: Publication date: Available online 9 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Wen Huang, Min Ji, Zhenxin Liu, Yingfei Yi
Stochastic stability of a compact invariant set of a finite dimensional, dissipative system is studied in our recent work “Concentration and limit behaviors of stationary measures” (Huang et al., 2015) for general white noise perturbations. In particular, it is shown under some Lyapunov conditions that the global attractor of the systems is always stable under general noise perturbations and any strong local attractor in it can be stabilized by a particular family of noise perturbations. Nevertheless, not much is known about the stochastic stability of an invariant measure in such a system. In this paper, we will study the issue of stochastic stability of invariant measures with respect to a finite dimensional, dissipative gradient system with potential function f . As we will show, a special property of such a system is that it is the set of equilibria which is stable under general noise perturbations and the set S f of global minimal points of f which is stable under additive noise perturbations. For stochastic stability of invariant measures in such a system, we will characterize two cases of f , one corresponding to the case of finite S f and the other one corresponding to the case when S f is of positive Lebesgue measure, such that either some combined Dirac measures or the normalized Lebesgue measure on S f is stable under additive noise perturbations. However, we will show by constructing an example that such measure stability can fail even in the simplest situation, i.e., in 1 -dimension there exists a potential function f such that S f consists of merely two points but no invariant measure of the corresponding gradient system is stable under additive noise perturbations. Crucial roles played by multiplicative and additive noise perturbations to the measure stability of a gradient system will also be discussed. In particular, the nature of instabilities of the normalized Lebesgue measure on S f under multiplicative noise perturbations will be exhibited by an example.
PubDate: 2015-10-11T17:52:05Z
- Abstract: Publication date: Available online 9 October 2015
- Radial symmetry on three-dimensional shells in the Landau-de Gennes theory
- Abstract: Publication date: Available online 9 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Giacomo Canevari, Mythily Ramaswamy, Apala Majumdar
We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landau-de Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature and (ii) for a fixed shell width at sufficiently low temperatures. In case (i), we provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution.
PubDate: 2015-10-11T17:52:05Z
- Abstract: Publication date: Available online 9 October 2015
- Stratification and enumeration of Boolean functions by canalizing depth
- Abstract: Publication date: Available online 8 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Qijun He, Matthew Macauley
Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively “picked off”, until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n -variable Boolean functions with depth k , which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.
PubDate: 2015-10-11T17:52:05Z
- Abstract: Publication date: Available online 8 October 2015
- Large-scale weakly nonlinear perturbations of convective magnetic dynamos
in a rotating layer- Abstract: Publication date: Available online 30 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): R. Chertovskih, V. Zheligovsky
We present a new mechanism for generation of large-scale magnetic field by thermal convection which does not involve the α -effect. We consider weakly nonlinear perturbations of space-periodic steady convective magnetic dynamos in a rotating layer of incompressible electrically conducting fluid that were identified in our previous work. The perturbations have a spatial scale in the horizontal direction that is much larger than the period of the perturbed convective magnetohydrodynamic state. Following the formalism of the multiscale stability theory, we have derived the system of amplitude equations governing the evolution of the leading terms in the expansion of the perturbations in power series in the scale ratio. This asymptotic analysis is more involved than in the cases considered earlier, because the kernel of the operator of linearisation has zero-mean neutral modes whose origin lies in the spatial invariance of the perturbed regime, the operator reduced on the generalised kernel has two Jordan normal form blocks of size two, and simplifying symmetries of the perturbed state are now missing. Numerical results for the amplitude equations show that a large-scale perturbation, periodic in slow horizontal variable, either converges to a short-scale neutral stability mode with amplitudes tending to constant values, or it blows up at a finite slow time.
PubDate: 2015-10-03T18:52:10Z
- Abstract: Publication date: Available online 30 September 2015
- Asymptotic analysis of a viscous thread extending under gravity
- Abstract: Publication date: Available online 21 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Jonathan J. Wylie, Huaxiong Huang, Robert M. Miura
Despite extensive research on extensional flows, there is no complete explanation for why highly viscous fluids extending under gravity can form such persistent and stable filaments with no sign of destabilization from surface tension. We therefore investigate the motion of a slender axisymmetric viscous thread that is supported at its top by a fixed horizontal surface and extends downward under gravity. In the case in which inertia and surface tension are initially negligible, we consider the long-wavelength equations for the full initial-boundary-value problem for a thread of arbitrary initial shape. We show that, eventually, the accelerations in the thread become sufficiently large that the inertial terms become important. Thus, we keep the inertial terms and, using matched asymptotic expansions, obtain solutions for the full initial-boundary-value problem. We show that the dynamics can be divided into two generic cases that exhibit very different behaviour. In the first case, the thread develops a long thin region that joins together two fluid masses. In this case, we use order-of-magnitude estimates to show that surface-tension-driven pinching will not occur if the square root of the Reynolds number is much greater than the initial aspect ratio divided by the Bond number. In the second case, the thread becomes thin near the horizontal surface. In this case, we show that the long-wavelength equations will ultimately break down and discuss the role of inertia in determining the dynamics. The asymptotic procedures require a number of novel techniques and the resulting solutions exhibit surprisingly rich behavior. The solution allows us to understand the mechanisms that underlie highly persistent filaments.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 21 September 2015
- The effects of wind and nonlinear damping on rogue waves and permanent
downshift- Abstract: Publication date: Available online 21 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): C.M. Schober, M. Strawn
In this paper we investigate the effects of wind and nonlinear damping on permanent downshift and the formation of rogue waves in the framework of a HONLS model. Wind effects are incorporated by including a uniform linear damping/forcing term in the model. The strength of the wind, Γ , is allowed to vary as well as wind duration. Determining permanent downshift is not straightforward and we propose a criteria for permanent downshift related to our numerical experiments. We consider large ensembles of initial data for modulated unstable Stokes waves with N = 1 , 2 , 3 unstable modes. In the nonlinear damped HONLS evolution we find permanent downshift is observed whenever the strength of the nonlinear damping β > 0.1 . Notably, rogue waves typically do not develop after the time of permanent downshift, implying that a downshifted sea-state does not allow for any further rogue waves. Incorporating wind effects into the nonlinear damped HONLS model, we find that damping by the wind weakens downshifting while forcing by the wind enhances downshifting. The proximity of the initial data to unstable plane waves impacts the characteristic features of the rogue waves in the nonlinear damped HONLS evolution. We find that as the initial data is chosen closer to the plane wave, the maximum strength, number, and lifetime of rogue waves increase while the time of permanent downshift decreases. Alternatively, we show that the greater the wave strength, the more rogue waves, or the longer their lifetime, the earlier permanent downshift occurs.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 21 September 2015
- Exact solutions of the Hirota equation and vortex filaments motion
- Abstract: Publication date: Available online 21 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): F. Demontis, G. Ortenzi, C. van der Mee
By using the Inverse Scattering Transform we construct an explicit soliton solution formula for the Hirota equation. The formula obtained allows one to get, as a particular case, the N -soliton solution, the breather solution and, most relevantly, a new class of solutions called multipole soliton solutions. We use these exact solutions to study the motion of a vortex filament in an incompressible Euler fluid with nonzero axial velocity.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 21 September 2015
- Singularity confinement and full-deautonomisation: A discrete
integrability criterion- Abstract: Publication date: Available online 14 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): B. Grammaticos, A. Ramani, R. Willox, T. Mase, J. Satsuma
We present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector. Our method is based on the full-deautonomisation procedure, which consists in analysing non-autonomous extensions of a given discrete system obtained by adding terms that are initially absent, but whose presence does not alter the singularity pattern. A justification for this approach is given through an algebro-geometric analysis. We also introduce the notions of early and late confinement. While the former is a confinement that may exist already for the autonomous system, the latter corresponds to a singularity pattern longer than that of the autonomous case. Late confinement will be shown to play an important role in the singularity analysis of systems with non-trivial gauge freedom, for which the existence of an undetected gauge in conjunction with a sketchy analysis, might lead to erroneous conclusions as to their integrability. An algebro-geometric analysis of the role of late confinement in this context is also offered. This novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping.
PubDate: 2015-09-18T07:48:29Z
- Abstract: Publication date: Available online 14 September 2015
- Predictability of threshold exceedances in dynamical systems
- Abstract: Publication date: Available online 10 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Tamás Bódai
In a low-order model of the general circulation of the atmosphere we examine the predictability of threshold exceedance events of certain observables. The likelihood of such binary events–the cornerstone also for the categoric (as opposed to probabilistic) prediction of threshold exceedences–is established from long time series of one or more observables of the same system. The prediction skill is measured by a summary index of the ROC curve that relates the hit- and false alarm rates. Our results for the examined systems suggest that exceedances of higher thresholds are more predictable; or in other words: rare large magnitude, i.e., extreme, events are more predictable than frequent typical events. We find this to hold provided that the bin size for binning time series data is optimized, but not necessarily otherwise. This can be viewed as a confirmation of a counterintuitive (and seemingly contrafactual) statement that was previously formulated for more simple autoregressive stochastic processes. However, we argue that for dynamical systems in general it may be typical only, but not universally true. We argue that when there is a sufficient amount of data depending on the precision of observation, the skill of a class of data-driven categoric predictions of threshold exceedences approximates the skill of the analogous model-driven prediction, assuming strictly no model errors. Therefore, stronger extremes in terms of higher threshold levels are more predictable both in case of data- and model-driven prediction. Furthermore, we show that a quantity commonly regarded as a measure of predictability, the finite-time maximal Lyapunov exponent, does not correspond directly to the ROC-based measure of prediction skill when they are viewed as functions of the prediction lead time and the threshold level. This points to the fact that even if the Lyapunov exponent as an intrinsic property of the system, measuring the instability of trajectories, determines predictability, it does that in a nontrivial manner.
PubDate: 2015-09-14T07:41:12Z
- Abstract: Publication date: Available online 10 September 2015
- Spectral transverse instabilities and soliton dynamics in the higher-order
multidimensional nonlinear Schrödinger equation- Abstract: Publication date: Available online 11 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Justin T. Cole, Ziad H. Musslimani
Spectral transverse instabilities of one-dimensional solitary wave solutions to the two-dimensional nonlinear Schrödinger (NLS) equation with fourth-order dispersion/diffraction subject to higher-dimensional perturbations are studied. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed using Fourier and finite difference differentiation matrices. It is found that for both signs of the higher-order dispersion coefficient there exists a finite band of unstable transverse modes. In the long wavelength limit we derive an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The time dynamics of a one-dimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations. Numerical nonlinear stability analysis is also addressed.
PubDate: 2015-09-14T07:41:12Z
- Abstract: Publication date: Available online 11 September 2015
- Experiments on a non-smoothly-forced oscillator
- Abstract: Publication date: Available online 9 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Lawrence N. Virgin, Christopher George, Ashwath Kini
This paper describes some typical behavior encountered in the response of a harmonically-excited mechanical system in which a severe nonlinearity occurs due to an impact. Although such systems have received considerable recent attention (most of it from a theoretical viewpoint), the system scrutinized in this paper also involves a discrete input of energy at the impact condition. That is, it is kicked when contact is made. One of the motivations for this work is related to a classic pinball machine in which a ball striking a bumper experiences a sudden impulse, introducing additional unpredictability to the motion of the ball. A one-dimensional analog of a pinball machine was the subject of a detailed mathematical study in Pring and Budd (2011), and the current paper details behavior obtained from a mechanical experiment and describes dynamics not observed in a conventional (passive) impact oscillator.
PubDate: 2015-09-10T07:33:26Z
- Abstract: Publication date: Available online 9 September 2015